Computationally-assisted multi-heterodyne spectroscopy

ABSTRACT

According to one aspect, a multi-heterodyne system is disclosed, which comprises a first laser source for generating multi-mode radiation having a frequency spectrum characterized by a first plurality of phase coherent frequencies, and a second laser source for generating multi-mode radiation having a frequency spectrum characterized by a second plurality of phase coherent frequencies. The system further comprises at least one detector for detecting a combination of the multi-mode radiation generated by the first and second laser sources so as to provide a multi-heterodyne signal having a frequency spectrum characterized by a plurality of beat frequencies, each beat frequency corresponding to a pairwise difference in the first and second plurality of phase coherent frequencies. The system further comprises an analyzer for receiving said multi-heterodyne signal and configured to employ a predictive model of the multi-heterodyne signal to provide estimates of any of phase error and timing error associated with the beat frequencies.

RELATED APPLICATION

The present application claims priority to provisional application No.62/216,417 filed on Sep. 10, 2015, which is herein incorporated byreference in its entirety.

GOVERNMENT SUPPORT

The invention was made with government support under NSF grant no.1505733 and DARPA contract no. W31P4Q-15-1-0009. Government has rightsin this invention.

BACKGROUND

The present invention relates generally to multi-heterodyne methods andsystems, and more particularly to methods and systems for correctingphase and timing errors in multi-heterodyne signals.

A frequency comb is a broadband coherent source whose frequency spectrumcan be fully described by two frequencies, namely, the offset and therepetition rate. Optical combs have found a variety of applications,e.g., in high precision metrology and spectroscopy. For example, in theterahertz frequency regime, combs generated by pulsed lasers can beuseful sources of radiation for detecting molecular finger-prints,because many molecules have strong rotational and vibrational resonancesin this frequency regime. Further, multi-heterodyne spectroscopy basedon two frequency combs, which is also known as dual-comb spectroscopy,allows performing broadband spectroscopy with a broad spectral coverage,a high frequency resolution, and high signal-to-noise ratios. Indual-comb spectroscopy, two frequency combs are directed onto a commondetector, and the heterodyne beating between different pairs of lines isdetected.

The implementation of dual-comb spectroscopy can be, however,challenging because the carrier-phase drift of the combs can precludecoherent averaging. If the drift is known, its effect can be corrected.But measuring the absolute frequency of a comb line can be challenging.One approach for measuring the carrier-envelope offset (CEO) of a combis to beat the comb with a stable continuous-wave (CW) laser. Anotherapproach is to use a narrowband optical filter, such as a Bragg grating,to select only a portion of a comb's optical spectrum, and to extractthe dual comb beating of different portions of the spectrum. Yet,another approach is to measure the CEO directly using a so-called f-2ftechnique. These conventional approaches, however, suffer from a numberof shortcomings. In particular, they can require the use of additionallasers and optical components, or can impose certain requirements on thecomb.

Moreover, performing dual-comb spectroscopy based on combs generated byquantum cascade lasers presents additional challenges. For example, theuse of reference channels in long wavelengths for phase and timingcorrection can require additional cryogenically cooled opticaldetectors. In addition, the lasers themselves are typicallycryogenically cooled in the long wavelength regime, and particularly inthe terahertz (THz) regime. Thus, the use of additional CW lasers inreference channels can greatly increase the cost and complexity of amulti-heterodyne system.

Accordingly, there is a need for improved multi-heterodyne methods andsystems, and more particularly, there is a need for improved methods andsystems for processing multi-heterodyne signals.

SUMMARY

According to one aspect, a multi-heterodyne system is disclosed, whichcomprises a first laser source for generating multi-mode radiationhaving a frequency spectrum characterized by a first plurality of phasecoherent frequencies, and a second laser source for generatingmulti-mode radiation having a frequency spectrum characterized by asecond plurality of phase coherent frequencies. The system furthercomprises at least one detector for detecting a combination of themulti-mode radiation generated by the first and second laser sources soas to provide a multi-heterodyne signal having a frequency spectrumcharacterized by a plurality of beat frequencies, each beat frequencycorresponding to a pairwise difference in the first and second pluralityof phase coherent frequencies. The system further comprises an analyzerfor receiving said multi-heterodyne signal and configured to employ apredictive model of the multi-heterodyne signal to provide estimates ofany of phase error and timing error associated with the beatfrequencies.

In some embodiments, the analyzer can correct any of the phase error andtiming error of the detected multi-heterodyne signal based on theestimates so as to generate a corrected multi-heterodyne signal. In someembodiments, the analyzer can further be configured to minimize an errorfunction associated with a difference between the detected and thepredicted multi-heterodyne signal to provide the estimated phase andtiming errors. In various embodiments, the any of an extended Kalmanfilter, an unscented Kalman filter, and a particle filter can be used tominimize the error function.

In one example, the predictive model of the multi-heterodyne signal(y(t)) may be defined as:

${y(t)} = {{\sum\limits_{n}{A_{n}e^{i{({\varphi_{0} + {n\;{\Delta\varphi}}})}}}} = {\sum\limits_{n}{r_{n}e^{i\;\varphi_{n}}e^{i\;{({\varphi_{0} + {n\;{\Delta\varphi}}})}}}}}$

wherein,

-   -   A_(n) denotes a complex amplitude associated with n^(th) beat        frequency characterized by a real amplitude r_(n) and a phase        φ_(n),

φ₀ denotes frequency offset phase between the multimode radiation fromsaid first and second lasers and is defined as follows:

${f_{0} = {\frac{1}{2\pi}\frac{d\;\varphi_{0}}{dt}}},$where ƒ₀ denotes a time-dependent frequency offset between two lowestfrequencies of the first and second plurality of frequencies,

Δφ denotes repetition rate phase and is defined as follows:

${{\Delta\; f} = {\frac{1}{2\pi}\frac{d\;{\Delta\varphi}}{dt}}},$where Δƒ denotes said repetition rate of said beat frequencies.In one example, the error function may be defined as:

${J(x)} = {{\sum\limits_{k}{{y_{k} - {h\left( x_{k} \right)}}}_{R^{- 1}}^{2}} + {{x_{k} - {f\left( x_{k - 1} \right)}}}_{Q^{- 1}}^{2}}$wherein,

-   -   x_(k) denotes a state of the system at time k,    -   y_(k) denotes measurement of the multi-heterodyne signal at time        k,    -   h(x_(k)) denotes measurement function h(x) evaluated at state        x_(k) as follows:    -   h(x_(k))=Σ_(n) A_(nk)e^(i(φ) ^(0k) ^(+nΔφ) ^(k)        ⁾=Σ_(n)r_(nk)e^(iφ) ^(nk) e^(i(φ) ^(0k) ^(+nΔφ) ^(k) ⁾, wherein        A_(nk), r_(nk), φ_(nk), φ_(0k), Δφ_(k) denote, respectively,        A_(n), r_(n), φ_(n), φ₀, and Δφ evaluated at time k,    -   ƒ(x_(k)) denotes time evolution function ƒ(x) evaluated at state        x_(k) such that:        r _(n(k+1)) =r _(nk)        φ_(n(k+1))=φ_(nk)        φ_(0(k+1))=φ_(0k)+2πΔt ƒ _(0k)        Δφ_(k+1)=Δφ_(k)+2πΔt Δƒ _(k)    -   wherein r_(n(k+1)), φ_(n(k+1)), φ_(0(k+1)), and Δφ_(k+1) denote,        respectively, r_(n), φ_(n), φ₀, and Δφ evaluated at time k+1,    -   R is said measurement noise covariance,    -   Q is said process noise covariance, such as noise associated        with the radiation sources, e.g., amplitude and phase noise.

In some embodiments, the frequency spectrum of any of the first andsecond pluralities of phase coherent frequencies can span a range of atleast about 1 octave.

In some embodiments, the system may further comprise an optical combinerfor receiving the radiation from the first and second lasers andgenerating a combined radiation beam directed to the detector. In someembodiments, at least one of the first and second lasers can generatecontinuous-wave (CW) radiation. In some embodiments, at least one of thefirst and second lasers can generate pulsed radiation. In someembodiments, at least one of the first and second lasers can generatechirped pulsed radiation. In some embodiments, at least one of the firstand second lasers can comprise a quantum cascade laser. In someembodiments, at least one of said first and second laser sources cancomprise an infrared laser source. In some embodiments, at least one ofsaid first and second laser sources can comprise a terahertz lasersource. In some embodiments, at least one of said first and second lasersources can comprise a laser diode. In some embodiments, at least one ofsaid laser sources can comprise a micro-ring resonator, e.g., amicro-ring resonator generating a frequency comb. In some embodiments,the multimode radiation generated by each of said first and second lasersources can comprise a frequency comb.

In some embodiments, the system can comprise two detectors, each ofwhich receives a combination of the multi-mode radiation generated bythe first and second laser sources to generate a multi-heterodynesignal. In some such systems, the analyzer can operate on themulti-heterodyne signal associated with one of the detectors to generatethe estimates of any of phase error and timing error and can apply thoseestimates to a respective multi-heterodyne signal generated by the otherdetector to generate a corrected multi-heterodyne signal.

In one example, the predictive model of said multi-heterodyne signal(y(t)) is defined as:

${y(t)} = {{\sum\limits_{n}{A_{n}e^{i\; 2\pi{\int{f_{n}{dt}}}}}} = {\sum\limits_{n}{r_{n}e^{i\;\varphi_{n}}e^{i\; 2\pi{\int{f_{n}d\; t}}}}}}$

wherein,

-   -   A_(n) denotes a complex amplitude associated with n^(th) beat        frequency characterized by a real amplitude r_(n) and a phase        φ_(n),    -   ƒ_(n) denotes the frequency of the n^(th) beat frequency.

In one example, the analyzer minimizes an error function defined as:

${J(x)} = {{\sum\limits_{k}{{y_{k} - {h\left( x_{k} \right)}}}_{R^{- 1}}^{2}} + {{x_{k} - {f\left( x_{k - 1} \right)}}}_{Q^{- 1}}^{2}}$

-   -   wherein,        -   x_(k) denotes a state of the system at time k,        -   y_(k) denotes measurement of the multi-heterodyne signal at            time k,        -   h(x_(k)) denotes the measurement function h evaluated at            state x_(k) and defined as follows:        -   h(x_(k))=Σ_(n) A_(nk)e^(i2πϕ) ^(nk) =Σ_(n)r_(nk)e^(iφ) ^(nk)            e^(iϕ) ^(nk) , wherein A_(nk), r_(nk), φ_(nk),        -   ϕ_(nk) denote, respectively, A_(n), r_(n), φ_(n) and ϕ_(n)            evaluated at time k,    -   ƒ(x_(k)) denotes the time evolution function ƒ evaluated at        state x_(k) such that:        r _(n(k+1)) =r _(nk)        φ_(n(k+1))=φ_(nk)        ϕ_(n(k+1))=ϕ_(nk)+2πΔt ƒ _(nk)    -   wherein r_(n(k+1)), φ_(n(k+1)), ϕ_(n(k+1)), denote,        respectively, r_(n), φ_(n), and    -   ϕ_(n) evaluated at time k+1,    -   R is said measurement noise covariance, and    -   Q is said process noise covariance, such as amplitude and noise        phase associated with the laser sources.

According to another aspect, a method for processing a multi-heterodynesignal is disclosed, which comprises generating from a first lasersource multi-mode radiation having a frequency spectrum characterized bya first plurality of phase coherent frequencies, and generating from asecond laser source multi-mode radiation having a frequency spectrumcharacterized by a second plurality of phase coherent frequencies. Themethod can further comprise detecting a combination of the multi-moderadiation generated by the first and second laser sources so as toprovide a multi-heterodyne signal having a frequency spectrumcharacterized by a plurality of beat frequencies, each beat frequencycorresponding to a pairwise difference between the first and secondplurality of phase coherent frequencies. The method can further comprisereceiving said multi-heterodyne signal and employing a predictive modelof said multi-heterodyne signal to provide estimates of any of phaseerror and timing error associated with the beat frequencies.

In some embodiments, the method can further comprise correcting any ofsaid phase error and timing error of the detected multi-heterodynesignal based on the estimates so as to generate a correctedmulti-heterodyne signal. The method can further comprise minimizing anerror function associated with a difference between the detected and thepredicted multi-heterodyne signal to provide the estimated phase andtiming errors. In one example, said error function may be minimizedusing an extended Kalman filter, an unscented Kalman filter, or aparticle filter.

In one example, the predictive model of said multi-heterodyne signal maybe defined as:

${y(t)} = {{\sum\limits_{n}{A_{n}e^{i\;{({\varphi_{0} + {n\;{\Delta\varphi}}})}}}} = {\sum\limits_{n}{r_{n}e^{i\;\varphi_{n}}e^{i\;{({\varphi_{0} + {n\;{\Delta\varphi}}})}}}}}$

wherein,

-   -   A_(n) denotes a complex amplitude associated with n^(th) beat        frequency characterized by a real amplitude r_(n) and a phase        φ_(n),    -   φ₀ denotes frequency offset phase and is defined as follows:

${f_{0} = {\frac{1}{2\pi}\frac{d\;\varphi_{0}}{dt}}},$where ƒ₀ denotes a time-dependent frequency offset between two lowestfrequencies of said first and second plurality of frequencies,

-   -   Δφ denotes repetition rate phase and is defined as follows:

${{\Delta\; f} = {\frac{1}{2\pi}\frac{d\;{\Delta\varphi}}{dt}}},$where Δƒ denotes said repetition rate of said beat frequencies.

In some embodiments, the method can further comprise combining themulti-mode radiation generated by said first and second lasers togenerate a combined beam for detection by the detector. In someembodiments, at least one of the first and second lasers can generatecontinuous-wave (CW) radiation. In some embodiments, at least one of thefirst and second laser sources can generate pulsed radiation. In someembodiments, at least one of the first and second laser sources cancomprise a quantum cascade laser. In some embodiments, at least one ofthe first and second laser sources can comprise an infrared lasersource. In some embodiments, at least one of the first and second lasersources can comprise a terahertz laser source.

According to a related aspect, a multi-heterodyne spectrometer isdisclosed, which comprises a first laser source for generatingmulti-mode radiation having a frequency spectrum characterized by afirst plurality of phase coherent frequencies, and a second laser sourcefor generating multi-mode radiation having a frequency spectrumcharacterized by a second plurality of phase coherent frequencies. Thespectrometer can further comprise a sample holder arranged such that themulti-mode radiation generated by at least one of the first and secondlaser sources passes through said sample holder so as to interact with asample contained therein. The spectrometer may further comprise at leastone detector for detecting a combination of the multimode radiationgenerated by the first and second lasers, wherein the combinationincludes at least one multimode radiation having passed through thesample holder, so as to generate a multi-heterodyne signal having afrequency spectrum characterized by a plurality of beat frequencies,each beat frequency corresponding to a pairwise difference between thefirst and second plurality of phase coherent frequencies. Thespectrometer can further comprise an analyzer for receiving themulti-heterodyne signal and configured to employ a predictive model ofthe multi-heterodyne signal to provide estimates of any of phase errorand timing error associated with said beat frequencies.

In some embodiments, the analyzer can correct any of said phase errorand timing error of the detected multi-heterodyne signal based on saidestimates so as to generate a corrected multi-heterodyne signal. In someembodiments, the analyzer can determine at least one property of thesample based on an analysis of said corrected multi-heterodyne signal.

In a related aspect, a method for processing a multi-heterodyne signalis disclosed, which comprises generating from a first laser sourcemulti-mode radiation having a frequency spectrum characterized by afirst plurality of phase coherent frequencies, generating from a secondlaser source multi-mode radiation having a frequency spectrumcharacterized by a second plurality of phase coherent frequencies,detecting a combination of said multi-mode radiation generated by saidfirst and second laser sources so as to provide a multi-heterodynesignal having a frequency spectrum characterized by a plurality of beatfrequencies, each beat frequency corresponding to a pairwise differencebetween said first and second plurality of phase coherent frequencies,and computationally estimating any of phase and timing error associatedwith said multi-heterodyne signal. In many embodiments, the step ofcomputationally estimating the phase and/or timing error is accomplishedby employing only the information that is contained within themulti-heterodyne signal itself. The error estimates can then be utilizedto compute a corrected multi-heterodyne signal.

Further understanding of various aspects of the invention can beobtained by reference to the following detailed description inconjunction with the associated drawings, which are described brieflybelow.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart illustrating an embodiment of a method ofprocessing a multi-heterodyne signal according to aspects of the presentdisclosure;

FIG. 2A schematically depicts two frequency-offset frequency combs;

FIG. 2B schematically depicts a plurality of beat frequencies generatedvia mixing the frequency combs depicted in FIG. 2A;

FIG. 3 is a schematic diagram of one embodiment of a multi-heterodynesystem according to aspects of the present disclosure;

FIG. 4 is a schematic diagram of an exemplary implementation of ananalyzer suitable for use in a multi-heterodyne system according toaspects of the present disclosure;

FIG. 5 is a schematic diagram of one embodiment of a multi-heterodynespectrometer according to aspects of the present disclosure;

FIG. 6 is a schematic diagram showing a hypothetical intensity variationof the beat frequencies associated with a corrected multi-heterodynesignal according to aspects of the present disclosure, which carriesinformation about a sample under study;

FIG. 7 is a schematic diagram of one embodiment of a spectrometeraccording to aspects of the present disclosure;

FIG. 8A depicts a multi-heterodyne signal detected via a hot electronbolometer (HEB) with a 100-μs acquisition time in a prototype devicebased on the embodiment of FIG. 7 with the effective noise floorindicated by a dashed line;

FIG. 8B depicts a multi-heterodyne signal detected by a Schottky mixerusing a prototype device based on the embodiment depicted in FIG. 7corresponding to, and detected during the same time period as, thesignal depicted in FIG. 8A;

FIGS. 8C and 8D show two multi-heterodyne teeth located, respectively,at 1788.5 MHz and 2472 MHz;

FIG. 9A shows multi-heterodyne signals generated with and without a GaAsetalon according to aspects of the present disclosure (the dashedhorizontal line indicates the threshold for inclusion in thetransmission data);

FIG. 9B shows measured etalon transmission and stimulated etalontransmission for the etalon associated with the multi-heterodyne signaldepicted in FIG. 9A;

FIG. 10A is a time domain signal of a combs' repetition rate measuredelectrically from a bias tree;

FIG. 10B shows chirping of repetition rates' difference associated withthe signal in FIG. 10A;

FIG. 10C is a frequency domain signal of combs' repetition rate measuredelectrically from a bias tee;

FIG. 10D is a time domain multi-heterodyne signal measured using an HEBdetector;

FIG. 10E shows chirping of the offset frequency associated with thesignal depicted in FIG. 10D in the interval depicted by the dashedlines,

FIG. 10F is a multi-heterodyne signal in the frequency domain, centeredat 900 MHz with 45 observable modes (the effective noise floor isindicated by a dashed line);

FIG. 11A schematically depicts an experimental set-up used to measure amulti-heterodyne signal generated via mixing of radiation from twomulti-mode lasers;

FIG. 11B shows quadrature components of a multi-heterodyne signalgenerated by source shown in FIG. 11A, obtained by IQ-demodulation;

FIG. 11C shows a frequency domain multi-heterodyne signal over anintegration time of 100 μs;

FIG. 11D shows frequency domain multi-heterodyne spectra obtained over 1μs at various times;

FIG. 12A shows repetition rate fluctuations of a demodulated RF comb;

FIG. 12B shows offset fluctuations of the demodulated RF comb;

FIGS. 12C and 12D show raw, predicted, and corrected multi-heterodynesignals during instability in the time domain signal and away from it;

FIG. 13A shows a raw multi-heterodyne signal;

FIG. 13B shows the signal in FIG. 13A following phase correction;

FIG. 13C shows the signal in FIG. 13A following phase and timingcorrection;

FIG. 14A shows plots of a simulated multiheterodyne signal, a simulatedcorrupted version of that signal due to phase and timing error, and asimulated restored version of that signal;

FIG. 14B shows a plot of estimated power of the simulatedmulti-heterodyne signal shown in FIG. 14A based on the restored signalrelative to the actual power used to generate the original simulatedsignal.

FIG. 14C shows a plot of residual error of the estimated power as afunction of actual power shown in FIG. 14B,

FIG. 14D shows a comparison of the timing error used to generate thecorrupted multi-heterodyne signal shown in FIG. 14A relative toestimates of the timing error obtained via analysis of the corruptedsignal; and

FIG. 14E shows a comparison of the phase error used to generate thecorrupted multi-heterodyne signal shown in FIG. 14A relative toestimates of the phase error obtained via analysis of the corruptedsignal.

DETAILED DESCRIPTION

The present invention relates generally to multi-heterodyne systems andmethods, and more particularly to methods and systems forcomputationally correcting any of the phase and timing error associatedwith a multi-heterodyne signal generated by beating two or more sets ofphase coherent frequencies by employing information contained within thesignal itself. Although in the following description, various aspects ofthe invention are described by reference to frequency combs, it shouldbe understood that the application of the present teachings are notlimited to frequency combs, but rather the present teachings can beapplied to any multi-heterodyne signal generated by beating two or moresets of phase coherent frequencies. As discussed in more detail below,in many embodiments, estimates of the phase and timing error associatedwith a measured multi-heterodyne signal are computationally estimatedand the estimates are employed to obtain a corrected multi-heterodynesignal. More specifically, in many embodiments, a measuredmulti-heterodyne signal and a predictive model of the signal areemployed in an error function and the error function is minimized toobtain estimates of the phase and the timing error. The presentteachings can be employed in connection with a variety of light sourcesthat generate phase coherent frequencies, and in particular frequencycombs. For example, frequency combs have been generated in the THz rangeby down-conversion of ultrafast laser pulses, which forms time-domainTHz pulses with well-defined phases. More recently, THz combs based onquantum cascade lasers have been generated via nonlinearities inlow-dispersion cavities.

Various terms are used herein consistent with their common meanings inthe art. By way of further illustration, the following terms as usedherein are defined as follows:

The term “frequency comb” as used herein refers to a frequency spectrumincluding a series of discrete, equally spaced frequencies.

The first-order coherence of a pair of oscillators of complex amplitudesA_(n) and A_(m) is defined as follows:

$\begin{matrix}{g_{n\; m} \equiv \frac{\left\langle {A_{n}^{*}A_{m}} \right\rangle}{\sqrt{{A_{n}}^{2}{A_{m}}^{2}}}} & (1)\end{matrix}$

The coherence of lines n and m is assessed by a heterodyne coherencemeasurement with any sufficiently fast detector having sufficientsensitivity to sense nonzero power of both line n and line m with asignal-to-noise ratio of at least 1 over some measurement time. Forexample, with a detector of noise equivalent power having NEP=10⁻⁹W/sqrt (Hz) and a line of power 1 μW, a sufficient integration timewould be roughly

${0.5\left( \frac{NEP}{P} \right)^{2}},$or about 0.5 μs.

The term “phase coherent frequencies” are used herein refers to at leasttwo frequencies (associated with frequency lines n and m) in thefrequency spectrum of a light source (typically a laser) that exhibit afirst-order coherence |g_(nm)|>0 within a 95% confidence level over themeasurement time defined by NEP to obtain a signal-to-noise ratio of atleast 1 for sensing the lines n and m.

The term “phase error” as used herein refers to phase fluctuations(e.g., phase drift) of one or more frequency lines present in afrequency spectrum, and more particularly, a frequency spectrumcomprising a plurality of phase coherent frequencies. With respect to amulti-heterodyne signal, phase error refers to phase fluctuations (e.g.,phase drift) associated with frequency lines in the signal, which can beobtained, e.g., by integrating frequency fluctuations over time. Thephase error in the multi-heterodyne signal can arise, for example, fromphase errors in the frequency combs generating the multi-heterodynesignal, which can in turn result, for example, from temperaturefluctuations, laser pump fluctuations, laser dynamics, mechanicalfluctuations, optical feedback, etc.

The term “timing error” as used herein refers to fluctuations in therepetition rate associated with a plurality of coherent frequencies(e.g., frequency comb). The timing error can result in fluctuations ofthe frequency spacing between adjacent frequency lines, which can inturn result, for example, from temperature fluctuations, laser pumpfluctuations, laser dynamics, mechanical fluctuations, optical feedback,etc.

With reference to the flow chart of FIG. 1, in one embodiments, a methodof processing a multi-heterodyne signal includes generating from a firstlaser source a multi-mode radiation having a frequency spectrumcharacterized by a first plurality of phase coherent frequencies (step1), and generating from a second laser source a multi-mode radiationhaving a frequency spectrum characterized by a second plurality of phasecoherent frequencies (step 2). A combination of the multi-mode radiationgenerated by the first and the second laser is detected to generate amulti-heterodyne signal having a frequency spectrum characterized by aplurality of beat frequencies, where each beat frequency corresponds toa pairwise frequency difference of the first and second plurality ofphase coherent frequencies (step 3). The detected multi-heterodynesignal and a predictive model of the multi-heterodyne signal areemployed to provide estimates of any of phase error and timing errorassociated with the beat frequencies (step 4). These estimates can thenbe utilized to correct any of the phase error and the timing errorassociated with the detected multi-heterodyne signal so as to produce acorrected multi-heterodyne signal (step 5).

The radiation generated by the first and the second laser can be acontinuous-wave (CW) radiation or a pulsed radiation. In someembodiments, the laser radiation generated by any of the first andsecond laser source can be a chirped pulsed radiation. The frequenciesassociated with the radiation generated by each laser can be in anysuitable region of the electromagnetic spectrum. For example, in someembodiments, the frequencies can be in the infrared region of theelectromagnetic spectrum (e.g., in a wavelength range of about 800 nm toabout 30 μm). In other embodiments, the frequencies can be in theterahertz region of the electromagnetic spectrum (e.g., in a range ofabout 300 GHZ to about 10 THz).

By way of illustration, FIG. 2A schematically depicts a first set offrequencies f1, f2, f3, f4, f5, and f6 associated with the radiationgenerated by the first laser and a second set of frequencies f′1, f′2,f′3, f′4, f′5 and f′6 associated with the second laser. In thisillustration, each of the first and the second frequency set is afrequency comb characterized by a plurality of equally spacedfrequencies. The frequencies of the first and the second comb are offsetrelative to one another.

More specifically, the first frequency comb (depicted with solid lines)has a mode spacing, also known as a repetition rate, ΔF₁, and the secondfrequency comb (depicted by broken lines) has a slightly differentrepetition rate, ΔF₂, where ΔF₂=ΔF₁+δ. By way of example, in someembodiments, each of the ΔF₁ and ΔF₂ can be in a range of about 1 GHz toabout 50 GHz.

As noted above, the detection of the combination of the first and secondmulti-mode radiation can result in the generation of a multi-heterodynesignal characterized by a plurality of beat frequencies (typically inthe radiofrequency (RF) regime) corresponding to the pairwise differenceof the frequencies present in the radiation generated by the two lasers.By way of illustration, FIG. 2B schematically depicts the beatfrequencies that can be present in a multi-heterodyne signal generatedvia mixing of the two frequency-offset frequency combs depicted in FIG.2A. For example, the beat frequencies RF1 and RF3 correspond,respectively, to the difference in the frequencies f1 and f′1 and f3 andf′3 of the two frequency combs. It should be understood that theseillustrative figures are provided only for ease of explanation ofvarious features of the invention, and not intended to limit the scopeof the present teachings. In particular, as noted above, theapplicability of the present teachings is not limited to frequencycombs.

Referring again to the flow chart of FIG. 1, the step of determiningestimates of the phase and/or the timing error can involve minimizing anerror function indicative of a difference between the detectedmulti-heterodyne signal and a predicted multi-heterodyne signalgenerated via the predictive model. A variety of different predictivemodels can be employed. For example, the predictive model of themulti-heterodyne signal can be defined in accordance with Equations (3)below. In some embodiments, the error function can also incorporate thestatistics of the noise associated the radiation generated by thelasers. By way of example, the amplitude noise and/or the phase noise ofone or more of the lasers can be taken into account.

After the phase and timing errors are estimated, the multiheterodynesignal can be corrected. For example, when Equation (3) below representsthe measured multiheterodyne signal (i.e., the signal isy(t)=Σ_(n)A_(n)e^(i(φ) ⁰ ^(+nΔφ))), phase correction is simple sincephase fluctuations are common to all comb lines and manifest as a puremultiplication. Therefore, phase correction can be performed bycalculating y₀(t)=e^(−iφ) ⁰ y(t). Timing fluctuations can present a moresubstantial challenge, since the effect of timing fluctuations is tononlinearly stretch the time axis. Defining an effective time

${{\tau(t)} = \frac{\Delta\varphi}{2\pi\left\langle {\Delta\; f} \right\rangle}},$the phase-corrected multiheterodyne signal can be written as y₀(t)=Σ_(n)A_(n)e^(in2π(Δƒ)τ). Therefore, the phase-timing correctedsignal can be found by interpolating y₀ (t) onto a uniform grid,effectively calculating y_(0Δ)(t)=y₀(τ⁻¹(t)). In one embodiment, astandard nonuniform Fast Fourier Transform can be Fused to perform theinterpolation.

The methods of the invention for processing a multi-heterodyne signalcan be implemented in a variety of systems. By way of example, FIG. 3schematically depicts an embodiment of such a system 10 having two lasersystems 12 and 14. Each of the laser systems 12 and 14 generates amulti-mode radiation having a frequency spectrum characterized by aplurality of phase coherent frequencies. More specifically, the laser 1generates a laser beam (LB1) having a frequency spectrum characterizedby a plurality of phase coherent frequencies, and the laser system 2generates a laser beam (LB2) having a frequency spectrum characterizedby a plurality of phase coherent frequencies. A variety of laser systemscan be employed. For example, in some embodiments one or both of thelasers 12 and 14 generate CW laser radiation while in other embodimentsone or both of the laser systems 12 and 14 can generate pulsedradiation. Further, the lasers 12 and 14 can operate in differentfrequency regimes, e.g., based on a desired application. For example,the lasers 12 and 14 can generate infrared, or terahertz radiation. Insome embodiments, the lasers can be diode lasers. In some embodiments,the lasers can be quantum cascade lasers, e.g., quantum cascade lasersoperating in the terahertz region of the electromagnetic spectrum.

With continued reference to FIG. 3, the system 10 further includes anoptical combiner 16 (e.g., a beam splitter) that receives the radiationbeam (LB1) generated from the laser 1 (via reflection of the beam from amirror 18) as well as the radiation beam (LB2) generated by the laser 2and overlaps the two radiation beams to generate a combined beam (LB3).In this embodiment, a convergent lens 20 focuses the combined beam ontoa detector 22, which detects the radiation to generate amulti-heterodyne detected signal. The detected signal includes afrequency spectrum characterized by a plurality of beat frequenciescorresponding to the differences between the frequencies associated withthe two laser beams. A plurality of different detectors can be employed,e.g., depending on the frequencies of the laser beams and/or aparticular application, among others. By way of example, the detector 22can be a hot electron bolometer detector, or a Schottky diode detector,among others.

The system 10 further includes an analyzer 24 that is electricallycoupled to the detector 22 so as to receive the multi-heterodyne signalgenerated by the detector. The analyzer can operate on themulti-heterodyne signal so as to provide estimates of the phase errorand/or timing error associated with the detected multi-heterodynesignal. By of example, the analyzer can employ the measuredmulti-heterodyne signal, e.g., over a given time interval, and apredictive model of the multi-heterodyne signal as input to an errorfunction and minimize the error function so as to obtain estimates ofthe phase error and/or the timing error. The analyzer can then employthose estimates to generate a corrected multi-heterodyne signal. Theanalyzer can employ a plurality of different error functions forobtaining estimates of the phase and/or the timing errors. By way ofexample, the error function an comprise a suitably regularized norm,such as ridge regression (also known as Tikhonov regularization), basispursuit (also known as LASSO), and the Dantzig Selector, etc. In someembodiments, unscented or scented Kalman filters can be used to minimizethe error function.

${{\tau(t)} \equiv \frac{{\Delta\varphi}(t)}{{2\pi} < {\Delta\; f} >}},$

By way of example, defining an effective time the multi-heterodynesignal y(t) can be written as y(t)=e^(iφ) ⁰^((t))(ΣnΣ_(n)A_(n)e^(in2π<Δƒ>τ(t)), where n refers to the frequencylines in the multi-heterodyne signal, A_(n) denotes the amplitude of then^(th) line. The phase corrected signal can be found by calculatingy₀(t)=e^(−iφ) ⁰ ^((t))y(t), and the phase-timing corrected signal can befound, for example, by using a non-uniform FFT (Fast Fourier Transform)to interpolate y₀(t) onto a linear grid, effectively calculatingy_(0Δ)(t)≡y₀(τ⁻¹(t)).

In some embodiments, an extended Kalman filter can be used forcomputationally obtaining estimates of the phase and the timing error.The extended Kalman filter can be viewed as fitting the measured data toa predictive model with a regularization constraint. Such anoptimization can be carried out, for example, by minimizing an errorfunction as follows:J(x)=Σ_(k) ∥y _(k) −h(x _(k))∥_(R) ⁻¹ ² +∥x _(k)−ƒ(x _(k−1))∥_(Q) ⁻¹²  (2)wherein,

x_(k) denotes a state of the system at time k,

y_(k) denotes measurement of the multi-heterodyne signal at time k,

h(x_(k)) denotes measurement function h(x) evaluated at state x_(k) asfollows:

-   -   h(x_(k))=Σ_(n) A_(nk)e^(i(φ) ^(0k) ^(+nΔφ) ^(k)        ⁾=Σ_(n)r_(nk)e^(iφ) ^(nk) e^(i(φ) ^(0k) ^(+nΔφ) ^(k) ⁾, wherein        A_(nk), r_(nk), φ_(nk), φ_(0k), Δφ_(k) denote, respectively,        A_(n), r_(n), φ_(n), φ₀, and Δφ evaluated at time k,        -   ƒ(x_(k)) denotes time evolution function ƒ(x) evaluated at            state x_(k) such that:            r _(n(k+1)) =r _(nk)            φ_(n(k+1))=φ_(nk)            φ_(0(k+1))=φ_(0k)+2πΔt ƒ _(0k)            Δφ_(k+1)=Δφ_(k)+2πΔt Δƒ _(k)        -   wherein r_(n(k+1)), φ_(n(k+1)), φ_(0(k+1)), and Δφ_(k+1)            denote, respectively, r_(n), φ_(n), φ₀, and Δφ evaluated at            time k+1,        -   R is said measurement noise covariance,        -   Q is said process noise covariance (such as amplitude and            noise of the laser sources), and ∥ν∥_(A) ²≡ν′Aν represents            the L₂-norm of ν with respect to the matrix A.

The first term in the above error function represents how closely thepredicted measurement matches the observed measurement, and the secondterm is a regularization term that takes the role of a time constant,controlling in this case how much multiplicative noise is present in thesystem.

In some embodiments, the following predictive model (herein alsoreferred to as the measurement model) can be utilized for modeling amulti-heterodyne signal (y(t)) can be defined as follows:y(t)=Σ_(n) A _(n) e ^(i(φ) ⁰ ^(+nΔφ))=Σ_(n) A _(n) e ^(iφ) ^(n) e ^(i(φ)⁰ ^(+nΔφ))  (3)

wherein,

-   -   A_(n) denotes a complex amplitude associated with n^(th) beat        frequency characterized by a real amplitude r_(n) and a phase        φ_(n),    -   φ₀ denotes frequency offset phase between the multimode        radiation from said first and second lasers and is defined as        follows:

${f_{0} = {\frac{1}{2\pi}\frac{d\;\varphi_{0}}{dt}}},$where ƒ₀ denotes a time-dependent frequency offset between two lowestfrequencies of said first and second plurality of frequencies,

Δφ denotes repetition rate phase and is defined as follows:

${{\Delta\; f} = {\frac{1}{2\pi}\frac{d\;{\Delta\varphi}}{dt}}},$where Δƒ denotes said repetition rate of said beat frequencies.

Alternative equivalent formulations of the predictive model also exist(e.g., considering the quadratures of the complex amplitudes). Assumingthat N is the number of frequency lines under consideration, the stateof the system can be described by a vector of length 2N+4, which wouldcontain the offset and the repetition rate (ƒ₀ and Δƒ), thecorresponding phases (φ₀ and Δφ)), the mode amplitudes (r_(n)), and themode phases (φ_(n)). At each timestep, the two frequencies, the modalamplitudes, and the phases can be assumed to be left substantiallyunchanged—perturbed only the Brownian noise—while the offset and timingphases can be updated by the frequencies:r _(n(k+1)) =r _(nk)φ_(n(k+1))=φ_(nk)φ_(0(k+1))=φ_(0k)+2πΔt ƒ _(0k)Δφ_(k+1)=Δφ_(k)+2πΔt Δƒ _(k)wherein r_(n(k+1)), φ_(n(k+1)), φ_(0(k+1)), and Δφ_(k+1) denote,respectively, r_(n), φ_(n), φ₀, and Δφ evaluated at time k+1.

Further, the process noise covariance, Q, can contain relatively largeamounts of phase and timing noise, relatively small amounts ofmultiplicative amplitude noise, and relatively small amounts ofadditional phase noise. In many embodiments, the additional phase noisecan be constructed in such a way that it would not contribute any extraphase/timing error, i.e., it can have rank N−2.

The above error function minimized by Kalman filter is nonconvex andconsequently possesses local minima. In some cases, the filter mayarrive at false minima (e.g., due to noise). Generally, two types ofsuch errors are possible: (1) the model comb's offset would lock to thetrue offset plus an integer multiple of the true repetition rate: ƒ₀^((model))=ƒ₀ ^((true))+nΔƒ^((true)), and

$\begin{matrix}{{\Delta\; f^{({model})}} = {\frac{n}{m}\Delta\;{f^{({true})}.}}} & (2)\end{matrix}$The first error is fairly trivial, and is in fact a consequence of thefact that the offset frequency of the RF comb is only defined modulo therepetition rate. The second error could in some cases affect the qualityof correction. In some embodiments, to remedy the effect of the seconderror, an estimate of the repetition rate is pre-calculated using thecoherence function C_(τ)(t)≡y⁺(t+τ)y(t), which contains frequencycomponents at Δƒ in addition to its harmonics. When the filter detectsthat the modeled repetition rate only has strong components at twolines—signifying that the model might be falsely locked—and is also farfrom the true repetition rate, the filter can then correct Δƒ bymultiplying by the appropriate rational number.

In some cases, a coherent artifact may be present in the extracted phaseand timing signals. For example, in some cases, if the process noise ofthe offset frequency is allowed to be large, the extracted offsetfrequency may contain spurious components at harmonics of the repetitionrate. In many cases, however, such components can be simply filtered outbecause the dual comb offset arises from the difference in the twocombs' individual offsets (which are generally unrelated).

The above equations (3) and (2) provide, respectively, examples of apredictive model and an error function that can be employed by theanalyzer 24 for estimating the phase error and/or the timing errorassociated with the multi-heterodyne signal, e.g., in a manner discussedabove.

The analyzer 24 can be implemented in hardware, firmware and/or softwareusing techniques known in the art and in accordance with the presentteachings. By way of example, FIG. 4 schematically depicts an exemplaryimplementation of analyzer 24, which includes an analog-to-digitalconverter 26 for receiving the detected multi-heterodyne signal from thedetector 22 and digitizing that signal. The analyzer further includes acentral processing unit (CPU) 28 for controlling the operation of theanalysis module, including performing calculations and logic operations.The analyzer also includes ROM (read only memory) 30, RAM (random accessmemory) 32 and permanent memory 34. A communications bus 36 facilitatescommunication among various components of the analyzer, includingcommunications between the CPU 28 and other components. The memorymodules can be used to store instructions for analyzing themulti-heterodyne signal, e.g., in a manner discussed above. By way ofexample, in some embodiments, instructions for data analysis, e.g.,instructions for performing the above steps discussed in connection withFIG. 1, can be stored in the ROM 30. The CPU can employ instructionsstored in ROM 30 to operate on digitized multi-heterodyne data stored inRAM 32 to generate estimates of the phase and timing error associatedwith the multi-heterodyne signal and generate a correctedmulti-heterodyne signal. The CPU can effect the storage of the correctedmulti-heterodyne signal in permanent memory 34, e.g., in a database.

In some aspects, the present teachings relate to a multi-heterodynespectrometer for obtaining information, e.g., spectroscopic information,about a sample under study. FIG. 5 schematically depicts such aspectrometer 40 that includes two lasers 42 and 44, each of whichgenerates laser radiation having a frequency spectrum characterized by aplurality of phase coherent frequencies, as discussed above. Morespecifically, the laser 1 generates a laser beam (LB1) that is reflectedby a mirror 46 to reach an optical combiner 48. The laser 2 generates alaser beam (LB2), which propagates to a sample holder 50 in which asample under study can be contained. The sample holder 50 can include aninput face 52 through which the laser beam (LB2) enters the sampleholder and an exit face 54 through which at least a portion of the beamexits the sample holder after its passage therethrough. The laser beam(LB2) interacts with the sample within the sample holder as itpropagates through the sample holder. By way of example, in someembodiments, the laser beam (LB2) may excite one or more opticaltransitions of the sample and hence be at least partially absorbed bythe sample. In some embodiments, the laser beam (LB2) may be scatteredby the sample.

The laser beam (LB3) exiting the sample holder propagates to the opticalcombiner 48 and is combined with the laser beam (LB1) to form a combinedlaser beam (LB4) characterized by at least partial overlap of the laserbeams (LB1) and (LB3). A convergent lens 54 focuses the combined laserbeam onto the detector 56, which generates a multi-heterodyne signalhaving a frequency spectrum characterized by a plurality of beatfrequencies corresponding to the pairwise mixing of the offsetfrequencies in the multi-mode radiation generated by the lasers 1 and 2.In effect, the laser beam (LB1) functions as a local oscillator (LO) forthe down-conversion of the frequencies present in the spectrum of thelaser beam (LB2) and hence the beam (LB3) exiting the sample holder.

Similar to the above system 10, an analyzer 58 receives themulti-heterodyne signal generated by the detector 56 and operates onthat signal to estimate and correct the phase and the timing errorsassociated with the multi-heterodyne signal (i.e., associated with thefrequencies present in the multi-heterodyne signal) in a mannerdiscussed above.

In addition, the analyzer 58 can be configured to analyze the correctedmulti-heterodyne signal to extract information about the sample. By wayof example, FIG. 6 schematically depicts a hypothetical intensityvariation of the beat frequencies associated with the correctedmulti-heterodyne signal. By way of example, the variation of theintensities of the beat frequencies can be analyzed (e.g., with respectto a respective variation of the intensities of the beat frequencieswhen the beam (LB2) passes through a sample holder without a sampletherein) to obtain information regarding the sample contained in thesample holder. For example, the decrease in the intensities of one ormore of the beat frequencies may be related to the absorption by thesample of the respective optical frequencies of the beam passing throughsample, thereby providing a spectroscopic signature of the sample.

The analyzer 58 can be implemented in a variety of ways using knowncomponents and techniques. For example, the analyzer 58 can beimplemented as shown in FIG. 4 and discussed above. In some embodiments,the instructions for correcting the multi-heterodyne signal receivedfrom the detector and for analyzing the corrected multi-heterodynesignal to extract information regarding the sample, e.g., itsspectroscopic signature, can be stored, e.g., in the permanent memory34. This information can be retrieved from the memory 34 by theprocessor 28 for the analysis of the multi-heterodyne signal receivedfrom the detector 56.

In some embodiments, the multi-heterodyne system can include twodetectors for detecting a multi-heterodyne signal generated via mixingof two (or more) frequency combs (or more generally two (or more)radiation beams characterized by a plurality of phase coherentfrequencies). In some such embodiments, one detector can function asreference detector for generating a multi-heterodyne signal that can beemployed to generate estimates of the phase and/or timing errors, whichcan then be applied to the multi-heterodyne signal generated by theother detector (herein also referred to as the measurement detector).

By way of illustration, FIG. 7 schematically shows such a system thatincludes two lasers 60 and 62, each of which generates a multi-modelaser radiation beam having a frequency spectrum characterized by aplurality of phase coherent frequencies. The radiation beam 64 generatedby the laser 60 is reflected by a curved reflector 66, which collimatesthe beam, toward a beam splitter 68, which reflects a portion of thebeam to a curved reflector 90 and allows the passage of the remainder ofthe beam toward another beam splitter 70. The laser beam 72 generated bythe laser 62 is in turn reflected by a curved reflector 74 to propagatetoward the beam splitter 76. A portion of the beam 62 is reflected bythe beam splitter 76 to pass through the beam splitter 68 to overlap theportion of the beam 64 that has been reflected by the beam splitter 68.The combined beam is focused by the curved reflector 90 onto a Schottkydiode detector 80. The Schottky diode detector generates amulti-heterodyne signal as a result of mixing of the two radiationbeams. An analyzer 82, similar to the analyzers discussed above, canreceive this multi-heterodyne signal and generate estimates of the phaseand timing errors.

With continued reference to FIG. 7, the portion of the laser beam 72that passes through the beam splitter 76 is reflected by a mirror 84 toreach a sample holder 86 in which a sample under study can be contained.After passage through the sample holder and interacting with the sample,the laser beam propagates to a beam splitter 70, which receives aportion of the other laser beam 64 after its passage through the beamsplitter 68 and combines the two beams to a combined beam that isfocused via a curved reflector 92 onto a hot electron bolometer detector88. The detector 88 generates a multi-heterodyne signal and the signalis received by the analyzer 82. The analyzer 82 employs the estimates ofthe phase error and timing error calculated based on the analysis of themulti-heterodyne signal generated by the detector 88 to generate acorrected multi-heterodyne signal. Further, the analyzer 82 can analyzethe multi-heterodyne signal to extract the sample's spectroscopicinformation, e.g., absorption, which is encoded in the multi-heterodynesignal.

The following examples are provided for further illustration of variousaspects of the invention. The examples are provided only forillustrative purposes and are not intended necessarily to indicate theoptimal ways of practicing the invention or the optimal results that maybe obtained.

Example 1

FIG. 7 illustrates one example of an experimental setup including twolasers 60 and 62, each laser generating a respective THz QCL comb thatwas used to generate the multi-heterodyne data. Both lasers werelens-coupled and had submilliwatt output powers at 37 K; when biasedinto the comb regime, their lasing spectra cover approximately 250 GHzat 2.8 THz. Further information regarding the structure of such THZ QCLlasers that are capable of generating radiation combs can be found,e.g., in an article entitled “Terahertz Laser Frequency Combs,” authoredby Burghoff et al. and published in Nature Photonics, vol. 8, June 2014,pp. 462-467, which is herein incorporated by reference in its entirety.In this example, to minimize their environmental differences, bothdevices were mounted inside the same pulsed-tube cryocooler 90. Toaccount for amplitude fluctuations, a balanced detection scheme wasemployed using one superconducting hot-electron bolometer (HEB) mixer 88that is helium-cooled as the signal detector, and one Schottky mixer 80operated at room temperature as the reference detector. In otherembodiments, other types of detectors may be used.

Both QCLs were biased into a comb regime and the repetition ratebeatnotes generated via mixing of their frequency combs were detectedusing a bias tee. The free-running combs featured repetition ratesaround 9.1 GHz and were separated by a 36 MHz difference, i.e., Δf 2−Δf1=36 MHz. At the same time, a multiheterodyne RF signal centered at 2.2GHz was detected from both the HEB and the Schottky mixer, indicatingthat these two combs' offset frequency differed by about 2.2 GHz. Themultiheterodyne signals were downconverted into the oscilloscope'sbandwidth by IQ demodulation with a synthesizer, and both the in-phaseand in-quadrature signals were then recorded with a fast oscilloscope.

The downconverted multiheterodyne signals were recorded for a durationof 100 μs and are shown in FIGS. 8A and 8B. Absolute power is expressedat the oscilloscope, and relative power is expressed with respect to thesystem's white noise. The signal from the HEB was used to generate aphase and timing correction signal, and this signal was used to correctboth interferograms. Because the radiation had the highly coherentstructure of a comb, only two frequency parameters were needed tocorrect all of the multiheterodyne lines. For example, FIGS. 8C and 8Dshow two multiheterodyne teeth from the Schottky mixer, located at1788.5 and 2472 MHz, which have full width half-maximum (FWHM)linewidths of 10.6 kHz and 2.6 kHz, respectively. Both linewidths wereat the Fourier uncertainty limit, implying that the correction procedurehad removed most of the phase and timing errors. The correctiveprocedure employed the predictive model and the error function presentedabove in Equations (3) and (2), respectively. The leftovermultiplicative noise after the phase and timing correction contributesto the noise floor of both multi-heterodyne signals, forming a noisepedestal indicated by dashed lines in FIGS. 8A and 8B.

With an acquisition time of 100 μs, the average SNR from the HEB wasabout 34 dB, and the apparent dynamic range (DNR) was about 52 dB. Themulti-heterodyne signal spanned 1.08 GHz with 30 distinguishable teeth,corresponding to optical spectrum coverage greater than 250 GHz at 2.8THz. The signal from the Schottky mixer had an average SNR of 24 dB anda DNR of 42 dB, although fewer lines were present than were visible fromthe HEB. The difference between the signals from the two detectorsmainly represents their differences in sensitivity, spectral response,and nonlinearity. In particular, saturation of the HEB generates severallines not present on the Schottky mixer, limiting its practical dynamicrange to about 37 dB. Still, both detectors are suitable for detectingstrong multiheterodyne signals.

Example 2

As a demonstration of broadband spectroscopy, transmission measurementsof a low-finesse etalon made from a tilted 625 μm thick undoped GaAswafer were performed. The signal and LO (local oscillator) lasers wereshined onto the HEB, and the etalon was placed in the signal laser'spath. For this measurement, no reference detector was used. FIG. 9Ashows the multi-heterodyne data collected from the HEB over 300 μs withand without the etalon, as shown by lines 300 and 302, respectively.FIG. 9B shows the ratio of individual multi-heterodyne peaks along withthe simulated transmission data (dotted line) at the frequencies thatwere sampled. To account for dynamic range limitations of the HEB, onlythose transmission values corresponding to the largest 24 lines, whosereference signal was greater than the peak intensity minus 37 dB, wereplotted. Periodic transmission due to the etalon was clearly visiblewithin the lasing spectral range, and is in reasonable agreement withthe theoretically calculated etalon transmission. Because no referencedetector was used, some residual errors were present on account ofrelative intensity fluctuations that occurred between the twomeasurements with and without the sample.

Example 3

Multi-heterodyne spectroscopy based on QCLs which are operated in pulsedbiasing mode (not to be confused with the optical pulses of amode-locked laser) was performed. It is well known that operating QCLsin continuous-wave mode is significantly more challenging than operatingthe same devices in pulsed mode, because CW operation places muchgreater thermal constraints on the laser in both the midinfrared and theterahertz. Many gain media simply have thresholds that are too high forCW operation and, even when CW operation is possible, the lasers' powerdissipation becomes problematic. For dual-comb THz spectroscopy, this isdoubly problematic because the two lasers are placed inside the samecooler. In addition, it is often desirable for spectroscopy to havesmall repetition rates, as the dense mode spacing eases the constraintson the detector and also makes it easier to achieve gapless coverage.This requires longer lasers that consume more power. As an example, 7 mmcombs were constructed, which consume approximately 1.3 A (about 1000A/cm²) and 15 V. Although these lasers have small free spectral ranges,around 4.8 GHz, the two of them together consume about 40 W. Thisconstitutes a major load on the cryocooler and would result in thelasers warming to above their maximum CW operating temperature.

FIGS. 10A-10F show the results of pulsed-mode, dual-comb spectroscopyusing the aforementioned devices. The lasers were biased to a combregime using 120 μs pulses with a repetition frequency of 100 Hz,resulting in a duty cycle of 1.2%. This low duty cycle significantlyeases the cryogenic operation. A low-pass filter was used to select onlythe part of the comb spectrum around 3.3 THz. FIGS. 10A and 10D show,respectively, in the time domain the combs' repetition rate signal(measured from a bias tee) and corresponding multi-heterodyne signal(measured from the HEB). As expected, both signals turn on during theelectrical pulse, but while the electrical repetition rate beatnotesturn on within a few microseconds, the optical multi-heterodyne signaltakes approximately 30 μs to stabilize. This reflects the fact thatelectrical beatnotes are unreliable indicators of optical beatnotes.FIG. 10C shows the distinct repetition rate beatnotes in the frequencydomain, clearly showing their frequency difference of 10 MHz. In pulsedmode, chirping of the repetition rate due to device heating isnoticeable; this heating results in a substantial broadening of thebeatnote indicated in FIG. 10C.

When the difference in their repetition rates is plotted in the timedomain, as shown in FIG. 10B, the chirp due to heating is evident.During the 50 μs period indicated by the two dashed lines, thedifference of the combs' repetition rates gradually increases from 10.1to 10.5 MHz. In addition, FIG. 10E shows the chirp of the offsetfrequency difference during the same 50 μs; it too is up-chirped.However, its magnitude is much larger, over 40 MHz, which isapproximately 100 times the repetition rate chirping. (It is noted thatquantifying the absolute frequency of each laser requires an absolutefrequency reference.) FIG. 10F shows the phase- and timing-correctedmulti-heterodyne signal in the frequency domain. The repetition ratedifference of 10 MHz is clearly visible here and over 45 modes arecontributing to the multi-heterodyne signal, implying a coverage of 215GHz in the THz spectrum. Within 50 μs of integration, the average SNR ofthe multi-heterodyne signal is higher than 25 dB on the HEB

Example 4

In this example, the lasers were heterogeneous QCLs that lase around 2.8THz and are dispersion-compensated, while the detectors used werehot-electron bolometers and Schottky-diode mixers. This example isfocused on coherent correction. FIG. 11A shows a simplified experimentalsetup along with the beatnotes of the lasers, labeled A and B. Thesebeatnotes are separated by 35 MHz, and should lead to an RF comb withrepetition rate of 35 MHz. However, because the lasers were operated ata bias in which they were only marginally stable, the beatnotesassociated with these lasers were quite broad. Additionally, laser Apossesses very clear sidebands (spaced by 140 kHz). As a consequence,the multi-heterodyne signal obtained from these devices, shown in thetime domain in FIG. 11B, is of poor quality and possesses very littleevidence of the periodicity that should arise from dual combs.Consequently, in the frequency domain, shown in FIG. 11C, themulti-heterodyne signal is broad and possesses very little evidence of adown-converted dual comb structure: over 100 s, the duration of therecording time, all features are completely washed out. One may viewthis as simply an issue of long term stability and that some structuremight be obtained by processing the signal over shorter time intervals;spectra over 1 s are shown in FIG. 11D. Though the comb structure is nowevident on some spectra, it is not the case for all of them. In fact,there remain many instances in which phase instabilities completelyspoil the spectrum, no matter how short the spectrum is cropped. As aresult, the signal needs to be corrected within the duration of aninterferogram by the instantaneous phase and timing signals.

Extracting the phase and timing errors from the observedmulti-heterodyne signal is essentially a nonlinear estimation problem.Even though there is no a priori knowledge of these errors, neverthelessthere is a model of what the RF comb should look like. Specifically, asdiscussed in detail above, the RF comb is expected to take the followingform:

${{y(t)} = {\sum\limits_{n}{A_{n}e^{{\mathbb{i}}{({\phi_{0} + {n\;{\Delta\phi}}})}}}}},$

where y(t) is the measured signal,

A_(n)=E*_(n,B)E_(n,A) is the dual comb amplitude of the nth line,

ϕ₀ and Δϕ are the phase corresponding to the offset and repetition ratesignals,

${f_{0,A} - f_{0,B}} = {{{\frac{1}{2\pi}\frac{d\;\phi_{0}}{dt}{\mspace{11mu}\;}{and}\mspace{14mu}\Delta\; f_{A}} - {\Delta\; f_{B}}} = {\frac{1}{2\pi}\frac{d\;\Delta\;\phi}{dt}}}$

In addition, the signal itself is corrupted by additive detector noise,and the parameters are all perturbed by multiplicative amplitude noiseand phase noise.

If the measurement was a linear function of the parameters, it would beexactly solvable by a Kalman filter. In the case of a nonlinearmeasurement one must linearize, resulting in an inexact solution.Nevertheless, good results can still be obtained. The process involvesfitting the measured multi-heterodyne signal to the dual comb model withthe constraint that the dual comb amplitudes vary slowly. Estimates ofthe offset and repetition rates may be continuously updated without anyform of cropping; this in principle makes it very amenable to real-timeprocessing. Alternatively, if the data has been recorded (as in thisexample), it is possible to perform RTS smoothing, using futureknowledge to refine the estimate and to correct for the group delayintroduced by the standard filter.

The physics of the comb enter primarily in the form of themultiplicative noise. Specifically, in this example, it is assumed thatthe comb complex amplitudes are perturbed only slightly at each timestep(giving them a long time constant), whereas the phase and timing errorsare perturbed much more (giving them a short time constant). In otherwords, it is assumed that the comb's phase noise covariance isapproximately rank-2. The Kalman filter quite naturally provides a wayto test the validity of this assumption, because at every timestep itmakes a prediction about what the next measurement will be. By simplycomparing the measured signal to the predicted signal, the accuracy ofthe prediction can be verified. For example, the prediction residual isunder 8% of the signal power.

FIGS. 12A and 12B show the instantaneous repetition rate and offsetfrequency of the RF comb discussed in connection with FIGS. 11A-D.Several features are immediately apparent. The first is that bothfrequencies suffer a perturbation that reoccurs every 7 μs, whichcorresponds to the aforementioned 140 kHz sidebands evident in thebeatnote of laser A. In other words, the beatnote undergoes a periodicinstability that is imprinted onto the multi-heterodyne spectrum.Secondly, the magnitude of the offset fluctuations (phase error) greatlyexceeds the magnitude of the repetition rate fluctuations (timingerror). This is not unexpected, since timing fluctuations correspondonly to group index whereas phase fluctuations also depend on phaseindex.

FIGS. 12C and 12D show the time-domain multi-heterodyne signals beforeand after the phase and timing correction, both during the instabilityand away from it. During the instability, no clear periodicity orstructure is obvious in the raw data; away from it, some periodicity isevident. In both cases, the signal predicted by the filter agrees verywell with the actual data. As a result, following the phase and timingcorrection, the periodic comb structure is recovered.

FIGS. 13A-13C show the results of the computational correction in thefrequency domain. The raw data from before is shown in FIG. 13A, andonce again shows no comb structure. The phase-corrected data is shown inFIG. 13B; because phase correction removes the average offset frequencyof the signal in addition to its fluctuations, the average offset <f0>has been re-added to correspond with the raw data. Phase correctionreveals the individual multi-heterodyne comb lines, although lines nearthe center of the comb are better-corrected than lines near the edgebecause of timing fluctuations are still present. FIG. 13C shows thephase- and timing-corrected spectra, with insets showing zoomed views ofseveral lines. All of the lines in the spectrum have been corrected,with full-width half maxima near the uncertainty limit of 10 kHz. Infact, following the correction some lines have appeared out of the noisefloor that were not apparent in the raw data, such as the one shown inthe rightmost inset. By filtering the data one can verify that this isreal signal (i.e., not a computational artifact), but given detectordynamic range limitations it may arise from detector nonlinearity ratherthan heterodyne beating. Even though the laser has a large disparity inmode amplitudes and large phase errors, with these techniques it ispossible to perform spectroscopy.

Systems and methods disclosed herein can deal with extremely largephase-timing fluctuations. For example, the laser may be biased in anunstable regime, causing the comb to chaotically switch between multipleoperating conditions. Even here, correction remains possible. As long asthe combs are coherent in the weak sense that the lines areevenly-spaced, with computational correction they become coherent in thestrong sense that mutually coherent dual comb spectroscopy can beperformed. Although various embodiments disclosed herein focus onunstable combs, the approach disclosed herein is also beneficial forstable combs and even combs operated in pulsed mode.

Although various examples disclosed herein use comb-like light sources,embodiments also work with light sources that are not comb-like, butmerely deterministic.

In addition, computational correction offers very good performance evenin the case of low signal reference measurements. For example,correction may be based on the multiheterodyne signal from a Schottkymixer, whose raw data has a signal-to-noise ratio (SNR) under 25 dB.Even when the mixer's noise is artificially boosted by 10 dB and littlesignal remains, computational correction remains informative on both thereference channel and a signal channel.

Example 5

In this example, demonstrated by FIGS. 14A, 14B, 14C, 14D, and 14E,simulated multiheterodyne data was generated, corrupted, and correctedso as to assess the efficacy of the correction. Artificial dual combdata was initially generated in such a way that the comb lines wouldhave power levels that were logarithmically-spaced, from the white noisepower level of 0 dB up to a maximum power level of 60 dB. The phases ofeach line and the order of the lines were chosen randomly. The spectrumis plotted in FIG. 14A and labeled “original.” Next, phase and timingerror were introduced by means of randomly-chosen noise signals, each ofwhich was generated by an autoregressive process with a characteristictime constant of 1 μs. The resulting multiheterodyne spectrum is alsoplotted in FIG. 14A and labeled as “corrupted;” the phase and timingcorruption result in a completely broadened spectrum. Finally,correction was performed using only information present in the broadenedspectrum as previously described; the original clean signal is recoveredas is plotted in FIG. 14A and labeled as “restored.”

In FIG. 14B, the power level of each line as determined by thecorrection procedure is plotted versus the actual power; good agreementis achieved between them (particularly at high signal levels). Byplotting the fractional residual error between each line as shown inFIG. 14C, it can be seen the largest signals have errors under 10⁻³,with the error increasing for lines that are closer to the noise floor.Lastly, FIGS. 14D and 14C show a comparison between the actual corruptedsignals with the estimated signals produced by the Kalman filterapproach. Once again, good agreement was found between them.

Those having ordinary skill in the art will appreciate that variouschanges can be made to the above embodiments without departing from thescope of the invention. All publications referenced herein are herebyincorporated by reference in their entirety.

What is claimed is:
 1. A multi-heterodyne system, comprising: a firstlaser source for generating multi-mode radiation having a frequencyspectrum characterized by a first plurality of phase coherentfrequencies, a second laser source for generating multi-mode radiationhaving a frequency spectrum characterized by a second plurality of phasecoherent frequencies, at least one detector for detecting a combinationof said multi-mode radiation generated by said first and second lasersources so as to provide a multi-heterodyne signal having a frequencyspectrum characterized by a plurality of beat frequencies each beatfrequency corresponding to a pairwise difference in said first andsecond plurality of phase coherent frequencies, an analyzer forreceiving said multi-heterodyne signal and configured to employ apredictive model of said multi-heterodyne signal to provide estimates ofany of phase error and timing error associated with said beatfrequencies.
 2. The system of claim 1, wherein said analyzer correctsany of said phase error and timing error of said detectedmulti-heterodyne signal based on said estimates so as to generate acorrected multi-heterodyne signal.
 3. The multi-heterodyne system ofclaim 2, wherein said predictive model of said multi-heterodyne signal(y(t)) is defined as:${y(t)} = {{\sum\limits_{n}{A_{n}e^{i\; 2\pi{\int{f_{n}d\; t}}}}} = {\sum\limits_{n}{r_{n}e^{i\;\varphi_{n}}e^{i\; 2\pi{\int{f_{n}{dt}}}}}}}$wherein, A_(n) denotes a complex amplitude associated with n^(th) beatfrequency characterized by a real amplitude r_(n) and a phase φ_(n),ƒ_(n) denotes the frequency of the n^(th) beat frequency.
 4. Themulti-heterodyne system of claim 3, wherein said analyzer minimizes anerror function defined as:${J(x)} = {{\sum\limits_{k}{{y_{k} - {h\left( x_{k} \right)}}}_{R^{- 1}}^{2}} + {{x_{k} - {f\left( x_{k - 1} \right)}}}_{Q^{- 1}}^{2}}$wherein, x_(k) denotes a state of the system at time k, y_(k) denotesmeasurement of the multi-heterodyne signal at time k, h(x_(k)) denotesthe measurement function h evaluated at state x_(k) and defined asfollows: h(x_(k))=Σ_(n) A_(nk)e^(i2πϕ) ^(nk) =Σ_(n)r_(nk)e^(iφ) ^(nk)e^(iϕ) ^(nk) , wherein A_(nk), r_(nk), φ_(nk), ϕ_(nk) denote,respectively, A_(n), r_(n), φ_(n) and ϕ_(n) evaluated at time k,ƒ(x_(k)) denotes the time evolution function ƒ evaluated at state x_(k)such that:r _(n(k+1)) =r _(nk)φ_(n(k+1))=φ_(nk)ϕ_(n(k+1))=ϕ_(nk)+2πΔt ƒ _(nk) wherein r_(n(k+1)), φ_(n(k+1)),ϕ_(n(k+1)), denote, respectively, r_(n), φ_(n), and ϕ_(n) evaluated attime k+1, R is said measurement noise covariance, and Q is said processnoise covariance.
 5. The multi-heterodyne system of claim 1, whereinsaid analyzer is further configured to minimize an error functionassociated with a difference between said detected and said predictedmulti-heterodyne signal to provide said estimated phase and timingerrors.
 6. The multi-heterodyne system of claim 5, wherein said errorfunction comprises any of an extended Kalman filter, an unscented Kalmanfilter, and a particle filter.
 7. The multi-heterodyne system of claim5, wherein said predictive model of said multi-heterodyne signal (y(t))is defined as:${y(t)} = {{\sum\limits_{n}{A_{n}e^{{i{({\varphi_{0} + {n\;{\Delta\varphi}}})}}\;}}} = {\sum\limits_{n}{r_{n}e^{i\;\varphi_{n}}e^{i{({\varphi_{0} + {n\;{\Delta\varphi}}})}}}}}$wherein, A_(n) denotes a complex amplitude associated with n^(th) beatfrequency characterized by a real amplitude r_(n) and a phase φ_(n), φ₀denotes frequency offset phase between the multimode radiation from saidfirst and second lasers and is defined as follows:${f_{0} = {\frac{1}{2\pi}\frac{d\;\varphi_{0}}{dt}}},$ where f₀ denotesa time-dependent frequency offset between two lowest frequencies of saidfirst and second plurality of frequencies, Δφ denotes repetition ratephase and is defined as follows:${{\Delta\; f} = {\frac{1}{2\pi}\frac{d\;{\Delta\varphi}}{dt}}},$ whereΔf denotes said repetition rate of said beat frequencies.
 8. Themulti-heterodyne system of claim 7, wherein said error function isdefined as:${J(x)} = {{\sum\limits_{k}{{y_{k} - {h\left( x_{k} \right)}}}_{R^{- 1}}^{2}} + {{x_{k} - {f\left( x_{k - 1} \right)}}}_{Q^{- 1}}^{2}}$wherein, x_(k) denotes a state of the system at time k, y_(k) denotesmeasurement of the multi-heterodyne signal at time k, h(x_(k)) denotesmeasurement function h(x) evaluated at state x_(k) as follows:h(x_(k))=Σ_(n) A_(nk)e^(i(φ) ^(0k) ^(+nΔφ) ^(k) ⁾=Σ_(n)r_(nk)e^(iφ)^(nk) e^(i(φ) ^(0k) ^(+nΔφ) ^(k) ⁾, wherein A_(nk), r_(nk), φ_(nk),φ_(0k), Δφ_(k) denote, respectively, A_(n), r_(n), φ_(n), φ₀, and Δφevaluated at time k, ƒ(x_(k)) denotes time evolution function ƒ(x)evaluated at state x_(k) such that:r _(n(k+1)) =r _(nk)φ_(n(k+1))=φ_(nk)φ_(0(k+1))=φ_(0k)+2πΔt ƒ _(0k)Δφ_(k+1)=Δφ_(k)+2πΔt Δƒ _(k) wherein r_(n(k+1)), φ_(n(k+1)), φ_(0(k+1)),and Δφ_(k+1) denote, respectively, r_(n), φ_(n), φ₀, and Δφ evaluated attime k+1, R is said measurement noise covariance, Q is said processnoise covariance.
 9. The multi-heterodyne system of claim 1, whereinsaid frequency spectrum of any of said first and second pluralities ofphase coherent frequencies spans a range of at least about 1 octave. 10.The multi-heterodyne system of claim 1, further comprising an opticalcombiner for receiving the radiation from said first and second lasersand generating a combined radiation beam directed to said at least onedetector.
 11. The multi-heterodyne system of claim 1, wherein at leastone of said first and second lasers generates continuous-wave (CW)radiation.
 12. The multi-heterodyne system of claim 1, wherein at leastone of said first and second lasers generates pulsed radiation.
 13. Themulti-heterodyne system of claim 12, wherein at least one of said firstand second lasers generates chirped pulsed radiation.
 14. Themulti-heterodyne system of claim 1, wherein at least one of said firstand second lasers comprises a quantum cascade laser.
 15. Themulti-heterodyne system of claim 1, wherein at least one of said firstand second laser sources comprises an infrared laser source.
 16. Themulti-heterodyne system of claim 1, wherein at least one of said firstand second laser sources comprise a terahertz laser source.
 17. Themulti-heterodyne system of claim 1, wherein at least one of said firstand second laser sources comprise a laser diode.
 18. Themulti-heterodyne system of claim 1, wherein the multimode radiationgenerated by each of said first and second laser sources comprises afrequency comb.
 19. The multi-heterodyne system of claim 1, wherein saidat least one detector comprises two detectors, each of said detectorreceiving a combination of the multi-mode radiation generated by saidfirst and second laser sources to generate a multi-heterodyne signal,and wherein said analyzer operates on the multi-heterodyne signalassociated with one of said detectors to generate said estimates of anyof phase error and timing error and applies said estimates tomulti-heterodyne signal generated by the other detector to generate acorrected multi-heterodyne signal.
 20. The multi-heterodyne system ofclaim 1, wherein at least one of said laser sources comprises amicro-ring resonator.
 21. The multi-heterodyne system of claim 20,wherein said micro-ring resonator generates a frequency comb.
 22. Amethod for processing a multi-heterodyne signal comprising: generatingfrom a first laser source multi-mode radiation having a frequencyspectrum characterized by a first plurality of phase coherentfrequencies, generating from a second laser source multi-mode radiationhaving a frequency spectrum characterized by a second plurality of phasecoherent frequencies, detecting a combination of said multi-moderadiation generated by said first and second laser sources so as toprovide a multi-heterodyne signal having a frequency spectrumcharacterized by a plurality of beat frequencies, each beat frequencycorresponding to a pairwise difference between said first and secondplurality of phase coherent frequencies, and employing a predictivemodel of said multi-heterodyne signal to provide estimates of any ofphase error and timing error associated with said beat frequencies. 23.The method of claim 22, further comprising correcting any of said phaseerror and timing error of said detected multi-heterodyne signal based onsaid estimates so as to generate a corrected multi-heterodyne signal.24. The method of claim 22, further comprising minimizing an errorfunction associated with a difference between said detected and saidpredicted multi-heterodyne signal to provide said estimated phase andtiming errors.
 25. The method of claim 24, further comprising using anyof an extended Kalman filter, an unscented Kalman filter, and a particlefilter to minimize the error function.
 26. The method of claim 22,wherein said predictive model of said multi-heterodyne signal is definedas:${y(t)} = {{\sum\limits_{n}{A_{n}e^{i{({\varphi_{0} + {n\;{\Delta\varphi}}})}}}} = {\sum\limits_{n}{r_{n}e^{i\;\varphi_{n}}e^{i\;{({\varphi_{0} + {n\;{\Delta\varphi}}})}}}}}$wherein, A_(n) denotes a complex amplitude associated with n^(th) beatfrequency characterized by a real amplitude r_(n) and a phase φ_(n), φ₀denotes frequency offset phase and is defined as follows:${f_{0} = {\frac{1}{2\pi}\frac{d\;\varphi_{0}}{dt}}},$ where f₀ denotesa time-dependent frequency offset between two lowest frequencies of saidfirst and second plurality of frequencies, Δφ denotes repetition ratephase and is defined as follows:${\Delta\; f\frac{1}{2\pi}\frac{d\;{\Delta\varphi}}{dt}},$ where λfdenotes said repetition rate of said beat frequencies.
 27. The method ofclaim 22, further comprising combining the multi-mode radiationgenerated by said first and second lasers to generate a combined beamfor detection by said detector.
 28. The method of claim 22, wherein atleast one of said first and second lasers generates continuous-wave (CW)radiation.
 29. The method of claim 22, wherein at least one of saidfirst and second laser sources generates pulsed radiation.
 30. Themethod of claim 22, wherein at least one of said first and second lasersources comprises a quantum cascade laser.
 31. The method of claim 22,wherein at least one of said first and second laser sources comprises ainfrared laser source.
 32. The method of claim 22, wherein at least oneof said first and second laser sources comprises a terahertz lasersource.
 33. A multi-heterodyne spectrometer comprising: a first lasersource for generating multi-mode radiation having a frequency spectrumcharacterized by a first plurality of phase coherent frequencies, asecond laser source for generating multi-mode radiation having afrequency spectrum characterized by a second plurality of phase coherentfrequencies, a sample holder arranged such that the multi-mode radiationgenerated by at least one of said first and second laser sources passesthrough said sample holder so as to interact with a sample containedtherein, at least one detector for detecting a combination of themultimode radiation generated by said first and second lasers, whereinthe combination includes at least one multimode radiation having passedthrough the sample holder, so as to generate a multi-heterodyne signalhaving a frequency spectrum characterized by a plurality of beatfrequencies, each beat frequency corresponding to a pairwise differencebetween said first and second plurality of phase coherent frequencies,and an analyzer for receiving said multi-heterodyne signal andconfigured to employ a predictive model of said multi-heterodyne signalto provide estimates of any of phase error and timing error associatedwith said beat frequencies.
 34. The multi-heterodyne spectrometer ofclaim 33, wherein said analyzer corrects any of said phase error andtiming error of said detected multi-heterodyne signal based on saidestimates so as to generate a corrected multi-heterodyne signal.
 35. Themulti-heterodyne spectrometer of claim 34, wherein said analyzerdetermines at least one property of said sample based on an analysis ofsaid corrected multi-heterodyne signal.